Non-Commutative Geometry, Multiscalars, and the Symbol Map

Starting from the concept of the universal exterior algebra in non-commutative differential geometry we construct differential forms on the quantum phase-space of an arbitrary system. They bear the same natural relationship to quantum dynamics which ordinary tensor fields have with respect to classical hamiltonian dynamics.Comment: 8 pages, late

On the degree of the polynomial defining a planar algebraic curves of constant width

In this paper, we consider a family of closed planar algebraic curves $\mathcal{C}$ which are given in parametrization form via a trigonometric polynomial $p$. When $\mathcal{C}$ is the boundary of a compact convex set, the polynomial $p$ represents the support function of this set. Our aim is to examine properties of the degree of the defining polynomial of this family of curves in terms of the degree of $p$. Thanks to the theory of elimination, we compute the total degree and the partial degrees of this polynomial, and we solve in addition a question raised by Rabinowitz in \cite{Rabi} on the lowest degree polynomial whose graph is a non-circular curve of constant width. Computations of partial degrees of the defining polynomial of algebraic surfaces of constant width are also provided in the same way.Comment: 13 page

The determinants of participation in microfinance and its impact on rural welfare: case study of the National Development Bank Botswana's Temo Bokamoso Lending programme in Kweneng District

This thesis investigates the determinants of participation in microfinance programmes and its impact on rural welfare in order to suggest improvements to the level of participation and the effectiveness of the programmes. The level of participation by rural smallholders in Botswana is much lower than expected in spite of the availability of schemes and their ease of access. The study uses one of National Development Bank Botswana's credit schemes in Kweneng District to investigate this problem. A field survey was carried out on 112 smallholder farmers, half of whom are scheme participants, to establish what factors affect the probability of their participation in the scheme. The study applied a logit model to determine which variables significantly affected the probability of participation in the credit scheme. The results indicated that some variables like gender and educational status had little effect on the probability of participation. A number of policy variables which include age, previous experience of credit use, respondents' perception of group collateral, distance from the National Development Bank, access to irrigation and total landholding size were found to have a significant effect. On the welfare effect, more than half of participants in the microfinance credit scheme indicated that their welfare improved as a direct result of the programme. As future research, it would be useful if the study could be extended to all districts of the country to enable the generalization of findings and provide valuable information for agricultural and poverty alleviation policy purposes

A Universal Formula for Deformation Quantization on K\"ahler Manifolds

We give an explicit local formula for any formal deformation quantization, with separation of variables, on a K\"ahler manifold. The formula is given in terms of differential operators, parametrized by acyclic combinatorial graphs.Comment: 20 pages, 8 figure

Can We Look at The Quantisation Rules as Constraints?

In this paper we explore the idea of looking at the Dirac quantisation conditions as $\hbar$-dependent constraints on the tangent bundle to phase-space. Starting from the path-integral version of classical mechanics and using the natural Poisson brackets structure present in the cotangent bundle to the tangent bundle of phase- space, we handle the above constraints using the standard theory of Dirac for constrained systems. The hope is to obtain, as total Hamiltonian, the Moyal operator of time-evolution and as Dirac brackets the Moyal ones. Unfortunately the program fails indicating that something is missing. We put forward at the end some ideas for future work which may overcome this failure.Comment: 4-pages, late

From Classical to Quantum Mechanics: "How to translate physical ideas into mathematical language"

In this paper, we investigate the connection between Classical and Quantum Mechanics by dividing Quantum Theory in two parts: - General Quantum Axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoint operators and so on) - Quantum Mechanics properly that specifies the Hilbert space, the Heisenberg rule, the free Hamiltonian... We show that General Quantum Axiomatics (up to a supplementary "axiom of classicity") can be used as a non-standard mathematical ground to formulate all the ideas and equations of ordinary Classical Statistical Mechanics. So the question of a "true quantization" with "h" must be seen as an independent problem not directly related with quantum formalism. Moreover, this non-standard formulation of Classical Mechanics exhibits a new kind of operation with no classical counterpart: this operation is related to the "quantization process", and we show why quantization physically depends on group theory (Galileo group). This analytical procedure of quantization replaces the "correspondence principle" (or canonical quantization) and allows to map Classical Mechanics into Quantum Mechanics, giving all operators of Quantum Mechanics and Schrodinger equation. Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We find also that this approach gives a natural semi-classical formalism: some exact quantum results are obtained only using classical-like formula. So this procedure has the nice property of enlightening in a more comprehensible way both logical and analytical connection between classical and quantum pictures.Comment: 47 page

Remark on Quantum Nambu Bracket

We give an explicit realization of quantum Nambu bracket via matrix of multi-index, which reduces in the continunm limit to the classical Nambu bracket.Comment: Latex, 5page

The damped harmonic oscillator in deformation quantization

We propose a new approach to the quantization of the damped harmonic oscillator in the framework of deformation quantization. The quantization is performed in the Schr\"{o}dinger picture by a star-product induced by a modified "Poisson bracket". We determine the eigenstates in the damped regime and compute the transition probability between states of the undamped harmonic oscillator after the system was submitted to dissipation.Comment: Plain LaTex file, 11 page